Inverse_Laplace_Review

Inverse_Laplace_Review - Chapter 2 The Laplace Transform...

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ZG2004/JC2007 1 Chapter 2 The Laplace Transform Objective: Mathematical preparation for the study of next chapters Contents: 2.1 Introduction 2.2 Complex Numbers and Harmonic Motion 2.3 Laplace Transformation 2.4 Inverse Laplace Transformation 2.5 Solving Linear Differential Eqs. by Laplace Transform
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ZG2004/JC2007 2 2.4 Inverse Laplace Transform Where: is called the Inverse Laplace Transform Operator . Definition [ ] ) ( ) ( 1 s F t f = L 1 L How to do it? If F(s) is simple use table of Laplace pairs • If F(s) is complicated Apply PFE first, then use table PFE? Partial Fraction Expansion ) ( ) ( ) ( ) ( 2 1 s F s F s F s F n + + + = Where Fi(s) is in simple form and can be found in table
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ZG2004/JC2007 3 Steps of PFE ) ( ) ( ) ( s D s N s F = Steps: 1. Make the order of N(s) less than that of D(s) 2. Factor D(s) into linear factors and/or quadratic factors 3 Expand out F(s) as ) ( ) )( ( ) ( ) ( ) ( ) ( 2 2 1 c bs as p s p s s N s D s N s F + + + + = = + + + + + + + + + = = ) ( ) ( ) ( ) ( ) ( ) ( 2 2 1 2 2 1 1 c bs as D s D p s A p s A s D s N s F 4 Apply either “cover-up” or “undetermined coefficients” method to find A1,A2, C1,D1…
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ZG2004/JC2007 4 PFE: Example of “Cover-Up’ Example 1: ) 5 ( 1 ) ( + = s s s F Solution: Find inverse Laplace 5 ) 5 ( 1 + + = + s B s A s s 1)Expand: 5 ) 5 ( + + = + s sB s sA s s s 3)Multiply by denominator, i.e. s 4) Cancel out common variables 2)Start with first coefficient A 5 ) 5 ( 1 + + = + s sB A s 5) Set denominator =0, i.e.
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Inverse_Laplace_Review - Chapter 2 The Laplace Transform...

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